While many discussions of relativity will discuss the increasing disparity between inertial frames of reference as one approaches the speed of light, one thing I've never seen anywhere, is what it is like AT the speed of light.

Since you cannot reach the speed of light through acceleration, we will have to imagine a hypothetical and basically supernatural situation where people could be "beamed" whole at the speed of light. (sort of like we imagine that matter could move faster than light; but only if it always moves faster than light. i.e "tachyons").

Extending it to the hypothetical limit of what happens as you approach the speed of light, the entire universe, in the direction you are traveling, flattens down to zero. The traveller basically sees himself passing through a "Flatland" having only height and width, and no depth. Its time is also frozen in one point (The point in which he was "emitted").

Really, because all three of his proper space dimensions are unchanged to him, and volume is h, w and d multiplied together, then the universe will be h × w × 0, which reduces the entire volume of the universe to zero. The universe basically doesn't exist from that frame of reference! Even though a photon can be deflected, changing direction, in our coordinate time; in its proper time, it has simply passed through an infinitely thin membrane. While photons do exist in our coordinate time, their "proper" frame of reference does NOT exist. It is flattened down to zero in the direction it is travelling.

Since the "forward" direction being travelled is still the same in proper time, yet the coordinate universe has been flattened down to zero in that same direction, then this "forward" dimension, (as well as "backward") comprises a new space dimension perpendicular to all three of the coordinate universe's dimensions! So one dimension of space has flattened down, and been replaced by another!.

In that direction, he will by default see himself at rest, just like all other observers.

In his frame of reference, a beam of light would still move forward at c. Coordinate time observers would "see" an infinitely flat ship (or whatever), frozen in time, moving at c.

However, in proper time, any light that moves past them at c, is really moving in a new dimension not shared by coordinate space, and the coordinate dimension the photon is travelling is perpendicular to all the dimensions it can measure.

This is interesting, as M-Theory now proposes a superspace consisting of a large fourth spatial dimension in addition to the six compacted ones. But, IIRC, it is only gravitons whose strings are free to move in this dimension; I don't think electromagnetism. (All other strings are believed to be attached to the three-brane we know as space, and thus limited to 3D). Still, since these hypothetical gravitons are also said to move at c, then perhaps that is the reason they have access to a large additional dimension.

That's how a paradox I noted was resolved. I had not before completed the thoughts on this, because of being stumped by what it would mean for c to still hold true in the proper time frame of reference of c, while still being c in our coordinate time. "Where" would the light be "going"? This explanation answers that!

Also, suspecting that if one space dimension begins flattening down as you approach c, then I wondered if time would then become a new space dimension, and would you see four space dimensions (h, w, and d and t partially flattened with one shrinking and the other unfurling) as you get closer to c? I couldn't make sense of it.

But then I remembered that you really can't accelerate to the speed of light. So as you are accelerating, you see the coordinate depth dimension shrink, while coordinate time slows down. All they'll do is continue to shrink, forever, down to fractions of spacelike units, but never reaching 0. You're not "getting [any] closer" to the speed of light, so you don't encounter any "inbetween" state with a partially extended extra dimension! You're always in your own proper spacetime, where the three dimensions plus time are always the same as they always were. It's the coordinate spacetime you see collapse, and the new dimension doesn't appear as such until the old space has collapsed completely to zero, and that only occurs AT the speed of light.

And again, it is really nothing more than your familiar back/forth dimension, which is now no longer part of coordinate space.

(If space is a hypersphere, you'll see it flatten down to a disk. If its infinite, it won't shrink at all; you'll only see the matter in it shrink, until you reach then end of matter's distribution. If it has some sort of "edge", beyond which there is no space, then there will be a shrinking distance you'll move in before you just cease to exist.

To be AT the speed of light is a quantum jump, in which you'll find one dimension completely collapsed, and a new one filling its place in the "ahead/behind" directions.

If you move in that dimension backwards, even close to c, you still do not come close to reentering the old dimension; you are still moving in the new dimension. All of coordinate space has now become the unapproachable c to you!

Likewise, if you move forward, you're not going faster than light. FTL was based on the coordinate frame of reference; and that no longer exists in proper space-time, remember. Your motion in the new dimension has no bearing on the old one.

Now, if space is a hypersphere, it will be an infinitely thin disk that you're perpetually "stuck" going through as you make an infinite number of laps in that direction, in zero proper time, and circumnavigating at 186,000m/s forever in coordinate time. If it has an edge, you'll instantly cease to exist. If its endless, then the distribution of matter (superclusters of galaxies; observable universe, etc) will be infinitely thin, but as for space itself; you'll have a 0×∞ paradox. (Another reason I don't believe in infinite space).

In any scenario, you'll be at the "lightlike infinity" at the boundaries of the Penrose diagram of the universe.

What about other c frames of reference traveling ahead or behind you? Like if you are "emitted" at one point in time, and then another c traveller is "emitted" one second behind you.

If you're traveling less than c, yet accelerating towards c, then in coordinate time, the distance between you approaches 186,000 miles. Yet the proper distance collapses towards zero. You in essence, would occupy the same proper space. However, the proper time approaches infinity! Remember, coordinate time grinds to a halt, so "one second behind you" increases towards forever.

You can see why, by graphing this on the Minkowski diagram. Coordinate space runs along at 0° to the right, while coordinate time is 90° (vertical), and c is 45°. However, proper time collapses towards 45° in both dimensions. As your timelike proper "here" heads outward at a decreasing angle, your spacelike proper "now" moves upward at an increasing angle.

At c, the space and time dimensions have collapsed into the c line at 45°.

So another c traveller emitted one second behind you will be a parallel 45° line that never intersects your line.

—at least not in that coordinate space direction being travelled.

Since the space between you decreases to nothing at the same time, this might balance itself out. How?

What I'm wondering, is that that new "forward" proper space dimension might be what represents coordinate "time". Thus fulfilling the theory that time and space are "exchanged" at or beyond the speed of light (and just like inside the Schwarzschild radius of a black hole).

So you might see your "follower" displaced (perhaps at 186,000 miles?) in that new "behind" dimension. If you both head backward approaching the speed of light in proper time, the distance would shrink, and proper time would also slow down. If you could make the "quantum jump" in that direction, you hypothetically would reemerge in coordinate time, I imagine at the point you made the quantum jump into the c frame.

This should give us an idea of FTL travel, in which coordinate/proper space and time become "imaginary" relative to each other.

Now, the collapsed space and time are not even zero; they are beyond zero. That is, in a multiplicative, not additive sense. It doesn't become simply "negative", which is the additive "beyond zero" realm. (Like take 1 and keep dividing it by finite positive numbers until you reach zero. And then imagine continuing to "past" zero. You can't even reach it in the first place). It is analogous to "greater than infinity". Thus, truly "imaginary".

In the FTL realm, the rules are the same. A beam of light will still pass you at c, but it will be a different dimension from the direction being traveled in coordinate space.

I believe that the so-called "tachyons" will thus not interact with coordinate spacetime at all (as has been speculated). Light speed is like the "bridge" between the sub- and super- luminous realms, where time and space are reversed like FTL, but you can still interact somewhat with the subluminous universe.

On the Minkowski diagram, while FTL's "here" (the actual world line) would lie in its expected spacelike (less than 45°) line; its "now", rather than lying in a timelike orientation, would generate a third dimension of the diagram; sticking out in the Z axis as a second space axis.